Abstract
Disordered fibrous networks are ubiquitous in nature as major structural components of living cells and tissues. The mechanical stability of networks generally depends on the degree of connectivity: only when the average number of connections between nodes exceeds the isostatic threshold are networks stable1. On increasing the connectivity through this point, such networks undergo a mechanical phase transition from a floppy to a rigid phase. However, even sub-isostatic networks become rigid when subjected to sufficiently large deformations. To study this strain-controlled transition, we perform a combination of computational modelling of fibre networks and experiments on networks of type I collagen fibres, which are crucial for the integrity of biological tissues. We show theoretically that the development of rigidity is characterized by a strain-controlled continuous phase transition with signatures of criticality. Our experiments demonstrate mechanical properties consistent with our model, including the predicted critical exponents. We show that the nonlinear mechanics of collagen networks can be quantitatively captured by the predictions of scaling theory for the strain-controlled critical behaviour over a wide range of network concentrations and strains up to failure of the material.
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References
Maxwell, J. C. On the calculation of the equilibrium and stiffness of frames. Phil. Mag. 27, 294–299 (1864).
Cates, M. E., Wittmer, J. P., Bouchaud, J. P. & Claudin, P. Jamming, force chains, and fragile matter. Phys. Rev. Lett. 81, 1841–1844 (1998).
Liu, A. J. & Nagel, S. R. Nonlinear dynamics: jamming is not just cool any more. Nature 396, 21–22 (1998).
Van Hecke, M. Jamming of soft particles: geometry, mechanics, scaling and isostaticity. J. Phys. Condens. Matter 22, 033101 (2010).
Thorpe, M. F. Continuous deformations in random networks. J. Non-Cryst. Solids 57, 355–370 (1983).
Jacobs, D. J. & Thorpe, M. F. Generic rigidity percolation: the pebble game. Phys. Rev. Lett. 75, 4051–4054 (1995).
Latva-Kokko, M., Mäkinen, J. & Timonen, J. Rigidity transition in two-dimensional random fiber networks. Phys. Rev. E 63, 046113 (2001).
Olsson, P. & Teitel, S. Critical scaling of shear viscosity at the jamming transition. Phys. Rev. Lett. 99, 178001 (2007).
Head, D. A. Critical scaling and aging in cooling systems near the jamming transition. Phys. Rev. Lett. 102, 138001 (2009).
Wyart, M., Liang, H., Kabla, A. & Mahadevan, L. Elasticity of floppy and stiff random networks. Phys. Rev. Lett. 101, 215501 (2008).
Ellenbroek, W. G., Zeravcic, Z., van Saarloos, W. & van Hecke, M. Non-affine response: jammed packings versus spring networks. Europhys. Lett. 87, 34004 (2009).
Broedersz, C. P., Mao, X., Lubensky, T. C. & MacKintosh, F. C. Criticality and isostaticity in fibre networks. Nature Phys. 7, 983–988 (2011).
Sheinman, M., Broedersz, C. P. & MacKintosh, F. C. Nonlinear effective-medium theory of disordered spring networks. Phys. Rev. E 85, 021801 (2012).
Lindström, S. B., Vader, D. A., Kulachenko, A. & Weitz, D. A. Biopolymer network geometries: characterization, regeneration, and elastic properties. Phys. Rev. E 82, 051905 (2010).
Licup, A. J. et al. Stress controls the mechanics of collagen networks. Proc. Natl Acad. Sci. USA 112, 9573–9578 (2015).
Head, D. A., Levine, A. J. & MacKintosh, F. C. Deformation of crosslinked semiflexible polymer networks. Phys. Rev. Lett. 91, 108102 (2003).
Wilhelm, J. & Frey, E. Elasticity of stiff polymer networks. Phys. Rev. Lett. 91, 108103 (2003).
Alexander, S. Amorphous solids: their structure, lattice dynamics and elasticity. Phys. Rep. 296, 65–236 (1998).
Fratzl, P. Collagen: Structure and Mechanics (Springer Science & Business Media, 2008).
Broedersz, C. P. & MacKintosh, F. C. Molecular motors stiffen non-affine semiflexible polymer networks. Soft Matter 7, 3186–3191 (2011).
Broedersz, C. P., Sheinman, M. & MacKintosh, F. C. Filament-length-controlled elasticity in 3d fiber networks. Phys. Rev. Lett. 108, 078102 (2012).
Conti, E. & MacKintosh, F. C. Cross-linked networks of stiff filaments exhibit negative normal stress. Phys. Rev. Lett. 102, 088102 (2009).
Straley, J. P. Critical phenomena in resistor networks. J. Phys. C 9, 783–795 (1976).
Achilli, M. & Mantovani, D. Tailoring mechanical properties of collagen-based scaffolds for vascular tissue engineering: the effects of ph, temperature and ionic strength on gelation. Polymers 2, 664–680 (2010).
Motte, S. & Kaufman, L. J. Strain stiffening in collagen I networks. Biopolymers 99, 35–46 (2013).
Arevalo, R. C., Kumar, P., Urbach, J. S. & Blair, D. L. Stress heterogeneities in sheared type-I collagen networks revealed by boundary stress microscopy. PLoS ONE 10, e011802 (2015).
Arrott, A. & Noakes, J. E. Approximate equation of state for nickel near its critical temperature. Phys. Rev. Lett. 19, 786–789 (1967).
Kane, C. & Lubensky, T. Topological boundary modes in isostatic lattices. Nature Phys. 10, 39–45 (2014).
Feng, J., Levine, H., Mao, X. & Sander, L. M. Alignment and nonlinear elasticity in biopolymer gels. Phys. Rev. E 91, 042710 (2015).
Arevalo, R. C., Urbach, J. S. & Blair, D. L. Size-dependent rheology of type-I collagen networks. Biophys. J. 99, L65–L67 (2010).
Head, D. A., Levine, A. J. & MacKintosh, F. C. Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletal and semiflexible polymer networks. Phys. Rev. E 68, 061907 (2003).
Acknowledgements
We thank M. Vahabi for useful discussions. This work was carried out on the Dutch national e-infrastructure with the support of SURF Cooperative. This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is financially supported by the Netherlands Organisation for Scientific Research (NWO). This work is further supported by NanoNextNL, a micro and nanotechnology programme of the Dutch Government and 130 partners.
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A.S., A.J.L. and K.A.J. contributed equally to the work. A.S., A.J.L., R.R., M.S. and F.C.M. conceived and developed the model and simulations. A.S., A.J.L., R.R. and M.S. performed the simulations. K.A.J. and G.H.K. designed the experiments. K.A.J. performed the experiments. All authors contributed to the writing of the paper.
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Sharma, A., Licup, A., Jansen, K. et al. Strain-controlled criticality governs the nonlinear mechanics of fibre networks. Nature Phys 12, 584–587 (2016). https://doi.org/10.1038/nphys3628
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DOI: https://doi.org/10.1038/nphys3628
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